Rust f32::asinh

Percentage Accurate: 38.0% → 99.5%

Time: 6.8s

Precision: binary32

Cost: 39528

• Unsound rule application detected in e-graph. Results may not be sound.

Math

\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

↓

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.5)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (+
         (* -0.16666666666666666 (pow x 3.0))
         (+ (* 0.075 (pow x 5.0)) (+ x (* -0.044642857142857144 (pow x 7.0)))))
        x)
       (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))))

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

↓

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.5f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + ((0.075f * powf(x, 5.0f)) + (x + (-0.044642857142857144f * powf(x, 7.0f))))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

↓

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.5))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(Float32(Float32(0.075) * (x ^ Float32(5.0))) + Float32(x + Float32(Float32(-0.044642857142857144) * (x ^ Float32(7.0)))))), x);
	else
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

↓

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.5))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + ((single(0.075) * (x ^ single(5.0))) + (x + (single(-0.044642857142857144) * (x ^ single(7.0)))))));
	else
		tmp = sign(x) * abs(log((abs(x) + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Target

Original	38.0%
Target	99.5%
Herbie	99.5%

\[\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}}\right), x\right) \]

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.5

   1. Initial program 49.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(x, x, 1\right) - x \cdot x\right) - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]
      Step-by-step derivation

      [Start] 49.4%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 49.4%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt{x \cdot x + 1} + \left|x\right|\right)}, x\right) \]
      add-sqr-sqrt [=>] -0.0%	\[ \mathsf{copysign}\left(\log \left(\sqrt{x \cdot x + 1} + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]
      fabs-sqr [=>] -0.0%	\[ \mathsf{copysign}\left(\log \left(\sqrt{x \cdot x + 1} + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]
      add-sqr-sqrt [<=] 7.9%	\[ \mathsf{copysign}\left(\log \left(\sqrt{x \cdot x + 1} + \color{blue}{x}\right), x\right) \]
      flip-+ [=>] 4.4%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot x}{\sqrt{x \cdot x + 1} - x}\right)}, x\right) \]
      log-div [=>] 4.4%	\[ \mathsf{copysign}\left(\color{blue}{\log \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot x\right) - \log \left(\sqrt{x \cdot x + 1} - x\right)}, x\right) \]
      add-sqr-sqrt [<=] 4.4%	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(x \cdot x + 1\right)} - x \cdot x\right) - \log \left(\sqrt{x \cdot x + 1} - x\right), x\right) \]
      fma-def [=>] 4.5%	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\mathsf{fma}\left(x, x, 1\right)} - x \cdot x\right) - \log \left(\sqrt{x \cdot x + 1} - x\right), x\right) \]
      +-commutative [=>] 4.5%	\[ \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x, 1\right) - x \cdot x\right) - \log \left(\sqrt{\color{blue}{1 + x \cdot x}} - x\right), x\right) \]
      hypot-1-def [=>] 4.5%	\[ \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x, 1\right) - x \cdot x\right) - \log \left(\color{blue}{\mathsf{hypot}\left(1, x\right)} - x\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]
      Step-by-step derivation

      [Start] 4.5%	\[ \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x, 1\right) - x \cdot x\right) - \log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right) \]
      fma-udef [=>] 4.4%	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(x \cdot x + 1\right)} - x \cdot x\right) - \log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right) \]
      +-commutative [<=] 4.4%	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x \cdot x\right)} - x \cdot x\right) - \log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right) \]
      associate--l+ [=>] 45.9%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x \cdot x - x \cdot x\right)\right)} - \log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right) \]
      +-inverses [=>] 100.0%	\[ \mathsf{copysign}\left(\log \left(1 + \color{blue}{0}\right) - \log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{copysign}\left(\log \color{blue}{1} - \log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{copysign}\left(\color{blue}{0} - \log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right) \]
      neg-sub0 [<=] 100.0%	\[ \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.200000003

   1. Initial program 19.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Applied egg-rr 20.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
      Step-by-step derivation

      [Start] 19.4%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      *-un-lft-identity [=>] 19.4%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]
      *-commutative [=>] 19.4%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]
      log-prod [=>] 19.4%	\[ \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]
      add-sqr-sqrt [=>] 10.1%	\[ \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      fabs-sqr [=>] 10.1%	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      add-sqr-sqrt [<=] 20.0%	\[ \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      +-commutative [=>] 20.0%	\[ \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]
      hypot-1-def [=>] 20.0%	\[ \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]
      metadata-eval [=>] 20.0%	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Simplified 20.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      Step-by-step derivation

      [Start] 20.0%	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0, x\right) \]
      +-rgt-identity [=>] 20.0%	\[ \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)}, x\right) \]

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 45.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Simplified 95.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Step-by-step derivation

      [Start] 45.9%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 45.9%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 95.6%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39528

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	39144

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	10120

\[\begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	97.3%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 6
Accuracy	99.4%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 7
Accuracy	96.7%
Cost	6760

\[\begin{array}{l} \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 8
Accuracy	79.8%
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9
Accuracy	95.8%
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 10
Accuracy	96.0%
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 11
Accuracy	64.4%
Cost	6532

\[\begin{array}{l} \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 12
Accuracy	62.6%
Cost	3400

\[\begin{array}{l} \mathbf{if}\;x \leq -5000:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \end{array} \]

Alternative 13
Accuracy	17.3%
Cost	3264

\[\mathsf{copysign}\left(-6, x\right) \]

Alternative 14
Accuracy	18.6%
Cost	3264

\[\mathsf{copysign}\left(15.333333333333334, x\right) \]

Reproduce

herbie shell --seed 2023165 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))